The results we show in this section are not optimal, they are optimal in the heavy-tailed setting but the effect due to corruption is sub-optimal compared to [1]}. One of our goals is to illustrate the use of the influence function and particularly Theorem REF  for an easy derivation of concentration inequalities for M-estimators, the other is to show that M-estimators are optimal in Heavy-tailed setting.
These examples also illustrate an interesting phenomenon derived from Theorem REF  by showing that the concentration of \(T(X_1^n)\)  around \(T(P)\)  can be much faster than the concentration of \(T(X_1^n)\)  around \(\mathbb {E}[X]\) , because the variance term is not \(Tr(\Sigma )\)  but \(V\)  which can be a lot smaller than \(Tr(\Sigma )\)  (for instance if \(Tr(\Sigma )\)  is not finite). It also shows that \(T(X_1^n)\)  is never arbitrarily bad as long as \(\mathbb {E}[\psi (\Vert X\Vert )]<\infty \)  contrary to the empirical mean.
We use the following corollary of [2]} recalled in Section , this lemma is proven in Section REF .
